Step 1:

If we substitute the limit into the function, we get:

[latex]\Large{\lim_{x\to -2}\frac{x^2-4}{x+2} = \frac{(-2)^2 \:- \:4}{(-2) \:+\: 2} = \frac{0}{0}}[/latex]

Of course, this is an indeterminate form, so we must change the function in some way, before substituting the limit.

Step 2:

Using the difference of squares rule, we can factor out the numerator, as shown:

[latex]\large{\lim_{x \to -2}\frac{x^2-4}{x+2} = \lim_{x\to -2}\frac{(x+2)(x-2)}{x+2}}[/latex]

[latex]\large{ = \lim_{x\to -2}(x-2)}[/latex]

Step 3:

Finally, we can plug in the limit again, to solve it:

[latex]\large{\lim_{x\to -2}(x-2) = (-2)\: -\: 2 = -4}[/latex]

This was a simple problem which involved evaluating limits by algebraic manipulation!

Solution Complete!